I got it to work by running cursor from WSL by running the command /mnt/c/Users/<windows username here>/AppData/Local/Programs/cursor/resources/app/bin/cursor /path/to/workspace/directory. This triggered the WSL server install, but on first load it complained that the WSL filesystem backend wasn’t present. I then hit ctrl-shift-p and ran the Extensions: Install Extensions command and saw that some extensions were requiring a window reload. Once I did that it seemed to be working.
Note that you may also need to add /mnt/c/Users/<windows username here>/AppData/Local/Programs/cursor/resources/app/bin to your PATH in WSL via the profile or rc script for your shell of choice if you want to just be able to run cursor . from within WSL.